Try to eat at home and buy food at «green» stores like VkusVill (ВкусВилл in Russian). Avoid city center for rentals. It is quite comfortable to get it by public transport. Cheapest retals are on the north. East is quite dangerous. Khimki — a nearby city on the north is also a good choice for budget in terms of rentals and ecology.

If you in the center you can use Market Place — it is good and not so expensive cafe with different food, or you can use #Pirogovii dvorik# or #teremok# — this is cafes with nice local food also not expensive and each of them have sets. For transport you can rent a bike (blue stations) and public transport. We have some special tickets for the tourists.

You can use an all day pass on the bus for 4,50$.
On food, the best and cheapest places are Burguer King, la lechonera, cuban café and supermarkets like Aldi, la chiquita or Fancy fruits.
For a place to stay in, i use Airbnb but if you don't feel comfortable using an app, you can check for hotels outside downtown disney and you will find rooms from 20$ to whatever the price you need.